多柔体系统数值分析的模型降噪方法
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摘要
多柔体系统的动力学方程通常是一组刚性微分方程,目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量,其求解效率难以令人满意.为了降低多柔体系统动力学方程的刚性,从而可采用ODE45等常规微分方程求解器进行求解,研究了在建模过程中滤除高频振荡分量的方法.在以当前时刻为起点的短时间内对柔性体的应力进行均匀化,用均匀化后的应力计算柔性体的变形虚功率,由此得到的系统动力学方程的解中不含过高频率的弹性振动,并且可以通过调节均匀化时间区间的长度参数控制滤波的范围.数值算例表明:这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器,并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率.本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.
Dynamic equations of flexible multibody systems are usually a set of stiff differential equations. At present, the common numerical method for solving the stiff differential equations filters out the high frequency by using the numerical damping. The computational efficiency of this method is still unsatisfactory. In order to reduce the stiffness of dynamic equations of flexible multibody systems so greatly that the equations can be solved by regular ordinary differential equation(ODE) solvers such as MATLAB ODE45 solver, methods of filtering high frequency vibrations during the process of modeling are studied. Stresses of flexible bodies are homogenized by their mean value over a time interval from now to a short time later. The homogenized stress is then employed to replace its origin when computing the virtual deformation power. In this way, the obtained model of the flexible multibody system will not contain harmful high frequency elastic vibrations. The range of frequencies can be controlled by the length of the time interval used to homogenize stresses. As validated by the numerical examples in this paper, the precision and efficiency of the proposed method are comparable to some stiff ODE solvers. Moreover, it works well when the stiff ODE solver fails to give correct solutions in a reasonable time. Comparisons of numerical examples show that the proposed method can be a new available approach to numerical analysis of flexible multibody systems.
Dynamic equations of flexible multibody systems are usually a set of stiff differential equations. At present, the common numerical method for solving the stiff differential equations filters out the high frequency by using the numerical damping. The computational efficiency of this method is still unsatisfactory. In order to reduce the stiffness of dynamic equations of flexible multibody systems so greatly that the equations can be solved by regular ordinary differential equation(ODE) solvers such as MATLAB ODE45 solver, methods of filtering high frequency vibrations during the process of modeling are studied. Stresses of flexible bodies are homogenized by their mean value over a time interval from now to a short time later. The homogenized stress is then employed to replace its origin when computing the virtual deformation power. In this way, the obtained model of the flexible multibody system will not contain harmful high frequency elastic vibrations. The range of frequencies can be controlled by the length of the time interval used to homogenize stresses. As validated by the numerical examples in this paper, the precision and efficiency of the proposed method are comparable to some stiff ODE solvers. Moreover, it works well when the stiff ODE solver fails to give correct solutions in a reasonable time. Comparisons of numerical examples show that the proposed method can be a new available approach to numerical analysis of flexible multibody systems.
引文
1田强,刘铖,李培等.多柔体系统动力学研究进展与挑战.动力学与控制学报,2017,15(5):385-405(Tian Qiang,Liu Cheng,Li Pei et al.Advances and challenges in dynamics of flexible multibody systems.Journal of Dynamics and Control,2017,15(5):385-405(in Chinese))
2王琪,庄方方,郭易圆等.非光滑多体系统动力学数值算法的研究进展.力学进展,2013,43(1):101-111(Wang Qi,Zhuang Fangfang,Guo Yiyuan,et al.Advances in the research on numerica methods for non-smooth dynamics of multibody systems.Advances in Mechanics,2013,43(1):101-111(in Chinese))
3刘才山,陈滨,彭瀚等.多体系统多点碰撞接触问题的数值求解方法.动力学与控制学报,2003,1(1):59-65(Liu Caishan Chen Bin,Peng Han,et al.Numerical resolution of multibody systems with multiple contact/impact points.Journal of Dynamics and Control,2003,1(1):59-65(in Chinese))
4姚文莉,岳嵘.有争议的碰撞恢复系数研究进展.振动与冲击,2015,34(19):43-48(Yao Wenli,Yue Rong.Advance in controversial restitution coefficient study for impact problems.Journal o Vibration and Shock,2015,34(19):43-48(in Chinese))
5 Stechlinski PG,Barton PI.Dependence of solutions of nonsmooth differential-algebraic equations on parameters.Journal of Differential Equations,2017,262(3):2254-2285
6 Dhamacharoen A.Efficient numerical methods for solving differential algebraic equations.Journal of Applied Mathematics&Physics2016,4(1):1-9
7 Petzold L.Differential/algebraic equations are not ODE’S.Siam Journal on Scientific&Statistical Computing,2012,3(3):367-384
8 Shampine LF,Reichelt MW,Kierzenka JA.Solving index-I DAEs in MATLAB and simulink.Siam Review,1999,41(3):538-552
9 Gear CW,Petzold LR.ODE methods for the solution of differential/algebraic systems.Siam Journal on Numerical Analysis,198421(4):716-728
10 Haddouni M,Acary V,Garreau S,et al.Comparison of severa formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics.Multibody System Dynamics,2017,41(3)201-231
11 Marques F,Souto A P,Flores P.On the constraints violation in forward dynamics of multibody systems.Multibody System Dynamics,2016,39(4):1-35
12 Schweizer B,Lu D,Li P.Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques.Multibody System Dynamics,2016,36(1):1-36
13潘振宽,赵维加,洪嘉振等.多体系统动力学微分/代数方程组数值方法.力学进展,1996,26(1):83-96(Pan Zhenkuan,Zhao Weijia,Hong Jiazhen,et al.On numerical algorithms for differential/algebraic equations of motion of multibody system.Advances in Mechanics,1996,26(1):26-40(in Chinese))
14袁新鼎.刚性常微分方程初值问题的数值解法.北京:科学出版社,1987(Yuan Xinding.Numerical Methods for Solving Initial Value Problems of Stiff Ordinary Differential Equations.Beijing:Science Press,1987(in Chinese))
15程正兴.数值逼近与常微分方程数值解.西安:西安交通大学出版社,2000(Cheng Zhengxing.Numerical Approximation and Numerical Solution of Ordinary Differential Equation.Xi’an:Xi’an Jiaotong University Press,2000(in Chinese))
16 Hairer E,Lubich C.Numerical Analysis of Ordinary Differential Equations.Berlin,Heidelberg:Springer 2015
17 Ibrahim Z,Nasir NM,Othman K,et al.Adaptive order of block backward differentiation formulas for stiff ODEs.Numerical Algebra,2017,7(1):95-106
18 El-Zahar ER,Habib HM,Rashidi MM,et al.A comparison of explicit semi-analytical numerical integration methods for solving stiff ODE systems.American Journal of Applied Sciences,2015,12(5):304-320
19 Ariel G,Engquist B,Tsai R.A multiscale method for highly oscillatory ordinary differential equations with resonance.Mathematics of Computation,2009,78(266):929-956
20 Tokman M.Efficient integration of large stiff systems of ODEs with exponential propagation iterative(EPI)methods.Journal of Computational Physics,2006,213(2):748-776
21 Meijaard JP.Application of Runge-Kutta-Rosenbrock methods to the analysis of flexible multibody systems.Multibody System Dynamics,2003,10(3):263-288
22 Shabana AA,Hussein BA.A two-loop sparse matrix numerical integration procedure for the solution of differential/algebraic equations:Application to multibody systems.Journal of Sound&Vibration,2009,327(3):557-563
23 Yen J,Petzold L,Raha S.A time integration algorithm for flexible mechanism dynamics:the DAEα-method.Computer Methods in Applied Mechanics and Engineering,1998,158(3):341-355
24 Arnold M,Brüls O.Convergence of the generalized-α,scheme for constrained mechanical systems.Multibody System Dynamics,2007,18(2):185-202
25丁洁玉,潘振宽.多体系统动力学微分?代数方程广义-α投影法.工程力学,2013,30(4):380-384(Ding Jieyu,Pan Zhenkuan generalized-αprojection method for differential-algebraic equations of multibody dynamics.Engineering Mechanics,2013,30(4)380-384(in Chinese))
26郭晛,章定国,陈思佳.Hilber-Hughes-Taylor-α法在接触约束多体系统动力学中的应用.物理学报,2017,66(16):144-154(Guo Xian,Zhang Dingguo,Chen Sijia.Application of Hilber-HughesTaylor-αmethod to dynamics of flexible multibody system with contact and constraint.Acta Physica Sinica,2017,66(16):144-154(in Chinese))
27姚廷强,迟毅林,黄亚宇.柔性多体系统动力学新型广义-α数值分析方法.机械工程学报,2009,45(10):53-60(Yao Tingqiang Chi Yilin,Hu Yayu.New generalized-αalgorithms for multibody dynamics.Journal of Mechanical Engineering,2009,45(10):53- 60(in Chinese))
28郭晛,章定国.柔性多体系统动力学典型数值积分方法的比较研究.南京理工大学学报(自然科学版),2016,40(6):726-733(Guo Xian,Zhang Dingguo.Comparative study of typical numerical integration methods of flexible multi-body systems dynamics.Journal of Nanjing University of Science and Technology,2016,40(6):726-733(in Chinese))
29 Banerjee A.Flexible Multibody Dynamics:Efficient Formulations and Applications.John Wiley&Sons,Inc.,2016
30洪嘉振.计算多体系统动力学.北京:高等教育出版社,1999(Hong Jiazhen.Computational Dynamics of Multibody Systems.Beijing:Higher Education Press,1999(in Chinese))
31齐朝晖.多体系统动力学.北京:科学出版社,2008(Qi Zhaohui.Dynamics of Multibody Systems.Beijing:Science Press,2008(in Chinese))
32张志刚,齐朝晖,吴志刚.一种基于应变插值大变形梁单元的刚?柔耦合动力学分析.振动工程学报,2015,28(3):337-344(Zhang Zhigang,Qi Zhaohui,Wu Zhigang.Rigid-flexible dynamics analysis of a large deformation beam element based on interpolation of strains.Journal of Vibration Engineering,2015,28(3):337-344(in Chinese))
2王琪,庄方方,郭易圆等.非光滑多体系统动力学数值算法的研究进展.力学进展,2013,43(1):101-111(Wang Qi,Zhuang Fangfang,Guo Yiyuan,et al.Advances in the research on numerica methods for non-smooth dynamics of multibody systems.Advances in Mechanics,2013,43(1):101-111(in Chinese))
3刘才山,陈滨,彭瀚等.多体系统多点碰撞接触问题的数值求解方法.动力学与控制学报,2003,1(1):59-65(Liu Caishan Chen Bin,Peng Han,et al.Numerical resolution of multibody systems with multiple contact/impact points.Journal of Dynamics and Control,2003,1(1):59-65(in Chinese))
4姚文莉,岳嵘.有争议的碰撞恢复系数研究进展.振动与冲击,2015,34(19):43-48(Yao Wenli,Yue Rong.Advance in controversial restitution coefficient study for impact problems.Journal o Vibration and Shock,2015,34(19):43-48(in Chinese))
5 Stechlinski PG,Barton PI.Dependence of solutions of nonsmooth differential-algebraic equations on parameters.Journal of Differential Equations,2017,262(3):2254-2285
6 Dhamacharoen A.Efficient numerical methods for solving differential algebraic equations.Journal of Applied Mathematics&Physics2016,4(1):1-9
7 Petzold L.Differential/algebraic equations are not ODE’S.Siam Journal on Scientific&Statistical Computing,2012,3(3):367-384
8 Shampine LF,Reichelt MW,Kierzenka JA.Solving index-I DAEs in MATLAB and simulink.Siam Review,1999,41(3):538-552
9 Gear CW,Petzold LR.ODE methods for the solution of differential/algebraic systems.Siam Journal on Numerical Analysis,198421(4):716-728
10 Haddouni M,Acary V,Garreau S,et al.Comparison of severa formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics.Multibody System Dynamics,2017,41(3)201-231
11 Marques F,Souto A P,Flores P.On the constraints violation in forward dynamics of multibody systems.Multibody System Dynamics,2016,39(4):1-35
12 Schweizer B,Lu D,Li P.Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques.Multibody System Dynamics,2016,36(1):1-36
13潘振宽,赵维加,洪嘉振等.多体系统动力学微分/代数方程组数值方法.力学进展,1996,26(1):83-96(Pan Zhenkuan,Zhao Weijia,Hong Jiazhen,et al.On numerical algorithms for differential/algebraic equations of motion of multibody system.Advances in Mechanics,1996,26(1):26-40(in Chinese))
14袁新鼎.刚性常微分方程初值问题的数值解法.北京:科学出版社,1987(Yuan Xinding.Numerical Methods for Solving Initial Value Problems of Stiff Ordinary Differential Equations.Beijing:Science Press,1987(in Chinese))
15程正兴.数值逼近与常微分方程数值解.西安:西安交通大学出版社,2000(Cheng Zhengxing.Numerical Approximation and Numerical Solution of Ordinary Differential Equation.Xi’an:Xi’an Jiaotong University Press,2000(in Chinese))
16 Hairer E,Lubich C.Numerical Analysis of Ordinary Differential Equations.Berlin,Heidelberg:Springer 2015
17 Ibrahim Z,Nasir NM,Othman K,et al.Adaptive order of block backward differentiation formulas for stiff ODEs.Numerical Algebra,2017,7(1):95-106
18 El-Zahar ER,Habib HM,Rashidi MM,et al.A comparison of explicit semi-analytical numerical integration methods for solving stiff ODE systems.American Journal of Applied Sciences,2015,12(5):304-320
19 Ariel G,Engquist B,Tsai R.A multiscale method for highly oscillatory ordinary differential equations with resonance.Mathematics of Computation,2009,78(266):929-956
20 Tokman M.Efficient integration of large stiff systems of ODEs with exponential propagation iterative(EPI)methods.Journal of Computational Physics,2006,213(2):748-776
21 Meijaard JP.Application of Runge-Kutta-Rosenbrock methods to the analysis of flexible multibody systems.Multibody System Dynamics,2003,10(3):263-288
22 Shabana AA,Hussein BA.A two-loop sparse matrix numerical integration procedure for the solution of differential/algebraic equations:Application to multibody systems.Journal of Sound&Vibration,2009,327(3):557-563
23 Yen J,Petzold L,Raha S.A time integration algorithm for flexible mechanism dynamics:the DAEα-method.Computer Methods in Applied Mechanics and Engineering,1998,158(3):341-355
24 Arnold M,Brüls O.Convergence of the generalized-α,scheme for constrained mechanical systems.Multibody System Dynamics,2007,18(2):185-202
25丁洁玉,潘振宽.多体系统动力学微分?代数方程广义-α投影法.工程力学,2013,30(4):380-384(Ding Jieyu,Pan Zhenkuan generalized-αprojection method for differential-algebraic equations of multibody dynamics.Engineering Mechanics,2013,30(4)380-384(in Chinese))
26郭晛,章定国,陈思佳.Hilber-Hughes-Taylor-α法在接触约束多体系统动力学中的应用.物理学报,2017,66(16):144-154(Guo Xian,Zhang Dingguo,Chen Sijia.Application of Hilber-HughesTaylor-αmethod to dynamics of flexible multibody system with contact and constraint.Acta Physica Sinica,2017,66(16):144-154(in Chinese))
27姚廷强,迟毅林,黄亚宇.柔性多体系统动力学新型广义-α数值分析方法.机械工程学报,2009,45(10):53-60(Yao Tingqiang Chi Yilin,Hu Yayu.New generalized-αalgorithms for multibody dynamics.Journal of Mechanical Engineering,2009,45(10):53- 60(in Chinese))
28郭晛,章定国.柔性多体系统动力学典型数值积分方法的比较研究.南京理工大学学报(自然科学版),2016,40(6):726-733(Guo Xian,Zhang Dingguo.Comparative study of typical numerical integration methods of flexible multi-body systems dynamics.Journal of Nanjing University of Science and Technology,2016,40(6):726-733(in Chinese))
29 Banerjee A.Flexible Multibody Dynamics:Efficient Formulations and Applications.John Wiley&Sons,Inc.,2016
30洪嘉振.计算多体系统动力学.北京:高等教育出版社,1999(Hong Jiazhen.Computational Dynamics of Multibody Systems.Beijing:Higher Education Press,1999(in Chinese))
31齐朝晖.多体系统动力学.北京:科学出版社,2008(Qi Zhaohui.Dynamics of Multibody Systems.Beijing:Science Press,2008(in Chinese))
32张志刚,齐朝晖,吴志刚.一种基于应变插值大变形梁单元的刚?柔耦合动力学分析.振动工程学报,2015,28(3):337-344(Zhang Zhigang,Qi Zhaohui,Wu Zhigang.Rigid-flexible dynamics analysis of a large deformation beam element based on interpolation of strains.Journal of Vibration Engineering,2015,28(3):337-344(in Chinese))